Hybrid waves for a (2 + 1)-dimensional extended shallow water wave equation

2021 ◽  
Vol 33 (11) ◽  
pp. 117120
Author(s):  
Gao-Fu Deng ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Ting-Ting Jia ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave ◽  
Hybrid Waves

scholarly journals Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

2021 ◽  
Author(s):  
Gao-Fu Deng ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Ting-Ting Jia ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Hybrid Solutions ◽  
The One ◽  
Hybrid Waves ◽  
Positive Integers

Abstract Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2+1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H -soliton, M -breather and J -lump solutions have been constructed via the modified Pfaffian technique, where H , M and J are the positive integers. One-breather solutions with a real function ϕ ( y ) are derived, where y is the scaled space variable, we notice that ϕ ( y ) influences the shapes of the background planes. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not affected by one soliton after interaction. One-lump solutions with ϕ ( y ) are obtained with the condition, where k 1 R and k 1 I are the real constants, we notice that the one lump consists of two low valleys and one high peak, as well as the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ ( y ) is changed from a linear function to a periodic function.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

Tanh-peakon and hyperbolicon of a nonlinear-strength shallow water wave equation

2007 ◽  
Vol 32 (2) ◽  
pp. 538-546 ◽  
Author(s):  
Lixin Tian ◽  
Min Fu
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave

scholarly journals On a shallow water wave equation

Nonlinearity ◽  
1994 ◽  
Vol 7 (3) ◽  
pp. 975-1000 ◽  
Author(s):  
P A Clarkson ◽  
E L Mansfield
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave

Exact Periodic Solitary Solutions to the Shallow Water Wave Equation

2009 ◽  
Vol 26 (5) ◽  
pp. 054701 ◽  
Author(s):  
Li Dong-Long ◽  
Zhao Jun-Xiao
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave ◽  
Solitary Solutions

scholarly journals Sharp local well-posedness for a fifth-order shallow water wave equation

2010 ◽  
Vol 369 (1) ◽  
pp. 133-143
Author(s):  
Wengu Chen ◽  
Zihua Guo ◽  
Zeping Liu
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Well Posedness ◽  
Shallow Water Wave ◽  
Fifth Order

scholarly journals The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ying Wang ◽  
YunXi Guo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Sufficient Conditions ◽  
Global Weak Solution ◽  
Dissipative Term ◽  
Shallow Water Wave ◽  
Global Strong Solution ◽  
Special Cases ◽  
The Cauchy Problem

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that(1-∂x2)u0∈M+(R),u0∈H1(R),andu0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.

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